Examples for the Yoneda lemma

Examples for the Yoneda lemma

[Hans-Peter Stricker]

Hello,

I am looking for (simple) instructive examples for the Yoneda lemma, showing
how to get the "inner" structure of an object from its morphisms. I've been
told how to get a graph G from its morphisms (from the one-vertex-graph V to
G and the one-edge-graph E to G and the morphisms from V to E) and
appreciated this example a lot. Are there others equally simple and
enlightening?

What I wonder is which morphisms are definitely needed. In the graph example
it's the morphisms from V -> G, E -> G and V -> E? Can this be abstracted
and generalized?

Many thanks in advance

Hans-Stricker




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Re: Examples for the Yoneda lemma

[Steve Lack]

Dear Hans-Peter,

In the case you mention, the category of graphs is a presheaf category (the
category of all functors C^op-->Set, for a small category C. The graphs V
and E you mention are precisely the representable functors, and the
morphisms V to E are the morphisms between these.

This is a general phenomenon: for any presheaf category [C^op,Set], you can
recover an object X when you know all the morphisms to X from the
representables, as well as how these morphisms behave when composed with
morphisms between representables.

The key property here is that the representables form a dense full
subcategory of [C^op,Set], and there is a further generalization to that
setting.

For another concrete example, consider the category Ab of abelian groups.
The elements of a group A can be identified with the morphisms Z-->A. Pairs
of elements can be identified with morphisms Z^2-->A, and the group
operation can then be recovered using the diagonal map Z-->Z^2. The other
operations can be recovered similarly (you'd also want to use the trivial
group in order to recover the unit element). If you also want to check the
associative law, you'd want to use the object Z^3 as well.

The connection between the previous two paragraphs is that the full
subcategory of Ab consisting of Z, Z^2, Z^3, and 1 is dense. (Actually, you
could just use Z and Z^2 for this.)

A similar analysis can be done for the models of any Lawvere theory in place
of Abelian groups.

Steve Lack.





On 15/01/10 11:24 AM, "Hans-Peter Stricker" <stricker@epublius.de> wrote:

> Hello,
>
> I am looking for (simple) instructive examples for the Yoneda lemma, showing
> how to get the "inner" structure of an object from its morphisms. I've been
> told how to get a graph G from its morphisms (from the one-vertex-graph V to
> G and the one-edge-graph E to G and the morphisms from V to E) and
> appreciated this example a lot. Are there others equally simple and
> enlightening?
>
> What I wonder is which morphisms are definitely needed. In the graph example
> it's the morphisms from V -> G, E -> G and V -> E? Can this be abstracted
> and generalized?
>
> Many thanks in advance
>
> Hans-Stricker
>

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Examples for the Yoneda lemma

[Vaughan Pratt]

Lots of examples in http://boole.stanford.edu/pub/yon.pdf .  It's an
18-page paper, yet already by page 2 there are six examples, none of
them the usual graph example.

In coming to grips with those examples it is *very* helpful to realize
that every algebraic theory including the well-known ones having at
least one constant or constant operation has a unary subtheory obtained
by fixing all but one argument of every nonzeroary operation in the
clone (which will be just projection if and only if all nonzeroary
operations are projections).  And while you don't need it on page 2,
further on it is helpful to realize that for every algebraic theory, its
models form a full subcategory of a presheaf category.

In order to reach a broader audience the paper was written for an
algebraic audience.  If you're more familiar with category language than
algebraic language a little adaptation will be needed.  Let me know if
translating back into category theory gives any trouble, this would be
helpful feedback to have.

Vaughan Pratt

Hans-Peter Stricker wrote:
> Hello,
>
> I am looking for (simple) instructive examples for the Yoneda lemma,
> showing
> how to get the "inner" structure of an object from its morphisms. I've been
> told how to get a graph G from its morphisms (from the one-vertex-graph
> V to
> G and the one-edge-graph E to G and the morphisms from V to E) and
> appreciated this example a lot. Are there others equally simple and
> enlightening?
>
> What I wonder is which morphisms are definitely needed. In the graph
> example
> it's the morphisms from V -> G, E -> G and V -> E? Can this be abstracted
> and generalized?
>
> Many thanks in advance
>
> Hans-Stricker


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Examples for the Yoneda lemma

[Aleks Kissinger]

The simplest example I can think of is posets. If you represent a
poset as a category (i.e. a category with at most one arrow from A->B
such that A->B and B->A implies A=B), then an object A is completely
determined by the set of arrows going in to it.

In this context, the Yoneda embedding is the familiar result that any
poset P embeds fully and faithfully in the powerset of P, ordered by
subset inclusion.


Aleks

On Fri, Jan 15, 2010 at 12:24 AM, Hans-Peter Stricker
<stricker@epublius.de> wrote:
> Hello,
>
> I am looking for (simple) instructive examples for the Yoneda lemma, showing
> how to get the "inner" structure of an object from its morphisms. I've been
> told how to get a graph G from its morphisms (from the one-vertex-graph V to
> G and the one-edge-graph E to G and the morphisms from V to E) and
> appreciated this example a lot. Are there others equally simple and
> enlightening?
>
> What I wonder is which morphisms are definitely needed. In the graph example
> it's the morphisms from V -> G, E -> G and V -> E? Can this be abstracted
> and generalized?
>
> Many thanks in advance
>
> Hans-Stricker
>

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Examples for the Yoneda lemma

[Hans-Peter Stricker]

Hello Aleks,

I am not quite what to think of the poset of unlabeled graphs without
isolated vertices with the relation of embeddability: I have the feeling,
that such a graph is NOT completely determined by its set of in-arrows (see
http://epublius.de/Fragment_of_the_category_of_unlabeled_graphs_without_isolated_vertices.pdf
to see what I mean, e.g. vertices 3 and 4 or vertices 7,8,9).

Do I miss something?

Best
Hans-Peter

----- Original Message -----
From: "Aleks Kissinger" <aleks0@gmail.com>
To: "Hans-Peter Stricker" <stricker@epublius.de>
Cc: <categories@mta.ca>
Sent: Friday, January 15, 2010 12:07 PM
Subject: Re: categories: Examples for the Yoneda lemma


> The simplest example I can think of is posets. If you represent a
> poset as a category (i.e. a category with at most one arrow from A->B
> such that A->B and B->A implies A=B), then an object A is completely
> determined by the set of arrows going in to it.
>
> In this context, the Yoneda embedding is the familiar result that any
> poset P embeds fully and faithfully in the powerset of P, ordered by
> subset inclusion.
>
>
> Aleks

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Examples for the Yoneda lemma

[Aleks Kissinger]

Here, it's important to distinguish graph and category. A (small)
category is a graph in some sense, but not all graphs are necessarily
categories. For instance, a category must have a composition operation
E x E -> E that is closed on edges.

As far as the poset-embedding example goes, Todd Trimble has an
excellent article on the topic:

http://topologicalmusings.wordpress.com/2008/04/02/toward-stone-duality-posets-and-meets/

Maybe this will clarify.


- Aleks

On Fri, Jan 15, 2010 at 12:50 PM, Hans-Peter Stricker
<stricker@epublius.de> wrote:
> Hello Aleks,
>
> I am not quite what to think of the poset of unlabeled graphs without
> isolated vertices with the relation of embeddability: I have the feeling,
> that such a graph is NOT completely determined by its set of in-arrows (see
> http://epublius.de/Fragment_of_the_category_of_unlabeled_graphs_without_isolated_vertices.pdf
> to see what I mean, e.g. vertices 3 and 4 or vertices 7,8,9).
>
> Do I miss something?
>
> Best
> Hans-Peter
>
> ----- Original Message ----- From: "Aleks Kissinger" <aleks0@gmail.com>
> To: "Hans-Peter Stricker" <stricker@epublius.de>
> Cc: <categories@mta.ca>
> Sent: Friday, January 15, 2010 12:07 PM
> Subject: Re: categories: Examples for the Yoneda lemma
>
>
>> The simplest example I can think of is posets. If you represent a
>> poset as a category (i.e. a category with at most one arrow from A->B
>> such that A->B and B->A implies A=B), then an object A is completely
>> determined by the set of arrows going in to it.
>>
>> In this context, the Yoneda embedding is the familiar result that any
>> poset P embeds fully and faithfully in the powerset of P, ordered by
>> subset inclusion.
>>
>>
>> Aleks
>>
>> On Fri, Jan 15, 2010 at 12:24 AM, Hans-Peter Stricker
>> <stricker@epublius.de> wrote:
>>>
>>> Hello,
>>>
>>> I am looking for (simple) instructive examples for the Yoneda lemma,
>>> showing
>>> how to get the "inner" structure of an object from its morphisms. I've
>>> been
>>> told how to get a graph G from its morphisms (from the one-vertex-graph V
>>> to
>>> G and the one-edge-graph E to G and the morphisms from V to E) and
>>> appreciated this example a lot. Are there others equally simple and
>>> enlightening?
>>>
>>> What I wonder is which morphisms are definitely needed. In the graph
>>> example
>>> it's the morphisms from V -> G, E -> G and V -> E? Can this be abstracted
>>> and generalized?
>>>
>>> Many thanks in advance
>>>
>>> Hans-Stricker
>>>
>>>
>>>
>>>
>>> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
>>>
>
>


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]